Applied Mathematics, 2011, 2, 830835 doi:10.4236/am.2011.27111 Published Online July 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Bayes Shrinkage Minimax Estimation in Inverse Gaussian Distribution Gyan Prakash Department of Community Medicine, Sarojini Naidu Medical College, Agra, India Email: ggyanji@yahoo.com Received November 19, 2010; revised May 14, 2011; accepted May 17, 2011 Abstract In present paper, the properties of the Bayes Shrinkage estimator is studied for the measure of dispersion of an inverse Gaussian model under the Minimax estimation criteria. Keywords: Bayes estimator, Bayes Shrinkage estimator, Uniformly Minimum Variance Unbiased Estimator (UMVUE), LINEX loss function (LLF) and Minimax Estimator 1. Introduction The Inverse Gaussian distribution plays an important role in Reliability theory and Life testing problems. It has useful applications in a wide variety of fields such as Biology, Economics, and Medicine. It is used as an im portant mathematical model for the analysis of positively skewed data. The review article by Folks & Chhikara [1,2] and Seshadri [3] have proposed many interesting properties and applications of this distribution. Let 12 be a random sample of size drawn from the inverse Gaussian distribution ,,,, n x x x,n μθIG , : having probability density function 2 32 ;, exp ; 22 0, ,0 θ xμ θ f x μθ π xμx x θ . (1) Here, stands for the mean and for the inverse measure of dispersion. The maximum likelihood esti mates of θ and θ are given as: 1 1 ˆi n i μ x x n andˆ , n θ v where 1 11 . i n i v xx The unbiased estimates of and θ are respect tively and 13 u θn v . Also, ~ IG, , μnθ 1n θ v ~ χ 2 with and being stochastically inde pendent ( [1,4,5,]). Schuster [6] showed that 2 2 1 ni ii x θ x μ is distributed as chi–square distribution with degrees of freedom. If we assume that n0 μ is known, the uniformly minimum variance unbiased (UMVU) estimator for measure of dispersion, 1 θ is 2 0 2 10 1ni ii xμ U nxμ (2) and follows a chisquare distribution with degrees of freedom. Un θn The choice of the loss function may be crucial. It has always been recognized that the most commonly used loss function, squared error loss function (SELF) is in appropriate in many situations. If the SELF is taken as a measure of inaccuracy then the resulting risk is often too sensitive to the assumptions about the behavior of the tail of the probability distribution. In addition, in some esti mation problems overestimation is more serious than the underestimation, or viceversa [7]. To deal with such cases, a useful and flexible class of asymmetric loss function (LINEX loss function (LLF)) was introduced by Varian [8]. The reparameterized version of LLF ([9]) for any parameter is given as θ ˆ 1; 0 and . a Le a a θθθ (3) The sign and magnitude of ‘a’ represents the direction and degree of asymmetry respectively. The positive (negative) value of ‘a’ is used when overestimation is v
G. PRAKASH 831 more (less) serious than underestimation. L is ap proximately square error and almost symmetric if a near to zero. Thompson [10] suggested a procedure, which makes use of a prior information of the parameter in form of a guessed value by shrinking the usual unbiased estimator towards the guess value of the parameter with the help of a shrinkage factor . The experimenter ac cording to his belief in the guess value specifies the val ues of shrinkage factor. The shrinkage estimator for the measure of dispersion of when a guess value of say is available, is given by 0k k 1 θIG 0, 1 . , μθ ,θ θ 1 0 ˆ1 ; 01 Sk θkθk (4) Some shrinkage estimators for measure of dispersion have been obtained by Pandey & Malik [11] and have studied their properties under SELF–criterion. Prakash and Singh [12] have studied the properties of different shrinkage testimators for under the LINEX loss function. Palmer [13] and Banerjee & Bhattacharya [14] have discussed the Bayesian inference about the parameters of the inverse Gaussian distribution. 1 θ 1 θ The present article proposed Bayes Shrinkage estima tor based on the Minimax criteria for the measure of dis persion. A Bayes estimator for the measure of dispersion under the vague prior has been obtained in the Sec tion 2. Under the Minimax criteria the Bayes Minimax estimator has been obtained in the Section 3. A Shrink age estimator construct by utilizing the Bayes Minimax estimator in the Section 4. Further, a numerical study has been presented in Section 5 and draws a conclusion about the Bayes Shrinkage Minimax estimator in Section 6. 1 θ 2. Bayes Estimator for Measure of Dispersion We are not going into debate or to justify the questions of the proper choice of the prior distribution. We con sider a vague prior for the parameter which is an increasing function of the parameter and is given as (); 0. d gθθd Therefore, the posterior density of parameter is defined as 22 ; π=; ; ;d nn U LUθg θ θLUθθe L Uθ g θθ After simplifying, the posterior density of parameter is obtained as 1 1 222 π1 22 n d nn θ d U nnU θΓ d θe . (5) The Bayes estimator for the measure of dispersion 1 θ ity under the LLF is obtained by simplifying the equal ˆ. a θθ a pp EθeeEθ Here, the suffix indicates that the expectation is taken under poensity. After simplification the Bayes estimator for p sterior d θ1 under the LLF is 2 ˆ; 1exp. 224 n a θU a n d (6) 3. The Minimax Bayes Estimator The basic principle of this approach is to minimize the n a theorem, n be stated as loss. The derivation depends primarily o given by Hodge & Lehmann [15] and ca follows. Let : θ ωFθ be a family of distribution func tions and C be a class of estimators of the parameter . Suppose that * cC is a Bayes estimator against a prior distribution θ on the parameter space g . Then the Bayes estimator * c is said to be the Minimax e timator if the riction of the estimator * c is in dependent on ssk fun . Here, the risk of the Bayes estimator given in (6) for the parameter 1 θ with respect to LLF is dened as fi 11Δ ˆ () a RθfU eˆ Δ1d ; Δ U a Uθθ θ (7) Since, nU is distributed as a Chisquare with n degrees of freedom. Then by making a transformn atio 2 nU z , the distribzution of is obtained as 11 n z n 2; 0 2 fz ezz (8) Using equation (8) in the expression we have ˆ Rθ 2 1 2 0 Γ 2 nn 2 12 ˆ1d 2 ˆ 111. a nz za n n a a Rθezeez az a Rθea n (9) The Equation (9) represents the risk of the Bayes esti mator of the measure of dispersion, which is independent with the parameter . Hence, the Bayes estimator is th y rent e Minimax estimator under the LLF loss criterion. The following statistical problem (Minimax Estima tion) is equivalent to some two person zero sum game between the Statistician (Player–II) and Nature (Plaer– I). Here the pure strategies of Nature are the diffe Copyright © 2011 SciRes. AM
G. PRAKASH 832 values of in the interval 0, and the mixed strategies of Nature are the prior densities of in the interval 0,. The pure strategies of Statistician are all possible decision functions in the interval 0,. The expected value of the loss function is the risk function and it is the gain of the Player–I. Furher, the Bayes risefined as ˆˆ t k is d , θ Rηθ ERθ Here, the expectation has been taken under the prior density of parameter . If the loss function is continu ous in both the estim and thearameterator ˆ θ p , and convex in for e of ˆ θeachvalu then therist m e ex easures * η and ˆ * θ for all and ˆ θ so that, the following relation holds: ˆˆˆ ,,, **** RηθRηθ Rηθ The number ˆ , ** Rηθ is known as th value of the game, and * η and ˆ * θ a e re the corresponding optimum strategies of tand II. In statisticas is the least fansity of h vora e Player I ble prior de l term* η and the estimtor for a ˆ* is the Miniator. In fact, the value of the game is the loss of the Player–II. Hence, the optimum strategy of Player–II and the value of game are given present case as Optimum Strategy Corresponding Loss Value of Game θmax estim ˆ θU LLF 2 2 1 n aa ea n 1 1 4. The Shrinkage Bayes Minimax Estimator Now, we construct a Shrinkage Bayes Minimax estima tor as 1 1. θUθ (10) 0 The risk of the Shrinkage Bayes Minimax estimator θ under the LLF is obtain by using Equation (8) as Δ Δ 1d 24 11 1 a RθfU eaU n d aδ (11) where U exp 1expan Rθ aδa 11 Δ θθθ and The comparison of the considnkage Bayes Minimax estimator 1 0. δθθ ered Shri θ is performs with the help of a minimum class of estimator based on the UMVU esti Here, the consideredof estimar based on U mator. class to MVU estimator is ; .TlUlR (12) The risk of the estimator T under the LINEX loss is given by 1.l 2 2 11 n aal RT ea n , which minimizes is given by The constantl RT 2 1exp . 22 na l an (13) ng the with the risk under the LLF is given as Thus, the improved estimators amo class T is TlU (14) 2 1exp1. n a RT a 22n (15) of Remark: It observed that the value rame um is lies be tween zero and one for the selected patric set of values which are considered later for the ne ings. Therefore, rical find is considered as the shrinkage factor. max estimator 5. A Numerical Study The relative efficiencies for the Shrinkage Bayes Mini θ relative to the improved estimator T is defined as , . RT RE θTRθ The relative efficiencies are the functions of and For the selected set of val ,n ues ,a of δ n .d 15,205,10, 5; 025,050; a. . ,01,150,02,05;. 025(..025)175. δ and 025d.,050. the relative efficiency e reha sen ve been calculated. The numerical findings arp ted hernly for 05n e o and 15n in the Tables It is obge Bayes Minimax esti mator 1 and 2 respectively. served that the Shrinka θ is performs better then the improved estimator T for the all selected parametr 025175.δ. ic set of values for . Furthmple size increases the re er, as san lative efficiency decreases for all considered paramet ric set values and attains maximum efficiency at the 1δ of point . Further, it is also observed that the relative eases as d increases when δ lie be tween 050 150.δ. efficiency incr . It is seen also that, as ‘a’ in crease relative efficiency first increases for 075δ. and thecrease for the other values of δ. 6. Con In present paper we obtained the Shrin ag n de clusions ke estimator Copyright © 2011 SciRes. AM
G. PRAKASH Copyright © 2011 SciRes. AM 833 ax estimation criteria for the measure of dispersion of the inverse Gaussian distribution. We ob based on the Minimserved that on the basis of the relative efficiency, the pro posed Shrinkage Bayes Minimax estimator θ performs θ Table 1. Relative efficiency for the estimator with respect toes . 0.25 * T under LLF for n = 5 and different valuof a, d and δ 0.5 1 1.5 2 a δd 0.25 1 1 1. 1..1782.15921127 1.06330161 0.50 1.6600 1.7066 1.7695 1.8010 1.8110 0.75 2.2234 2.4028 2.7503 3.0766 3.3768 1.00 2.5931 2.8804 3.5000 4.1796 4.9192 1.25 2.4673 2.6799 3.0899 3.4717 3.8190 1.50 1.9834 2.0465 2.1276 2.1635 2.1698 –0.05 –0.25 0.25 0.05 1.75 1.4748 1.4525 1.3928 1.3276 1.2655 0.25 1.1923 1.1737 1.1282 1.0797 1.0332 0.50 1.6869 1.7333 1.7953 1.8258 1.8348 0.75 2.2582 2.4388 2.7880 3.1147 3.4144 1.00 2.5904 2.8733 3.4829 4.1511 4.8778 1.25 2.3821 2.5787 2.9575 3.3106 3.6324 1.50 1.8466 1.9002 1.9698 2.0017 2.0086 1.75 1.3356 1.3150 1.2628 1.2069 1.1534 0.25 1.2177 1.2002 1.1567 1.1102 1.0655 0.50 1.7353 1.7815 1.8422 1.8710 1.8782 0.75 2.3158 2.4996 2.8530 3.1816 3.4808 1.00 2.5583 2.8327 3.4233 4.0697 4.7719 1.25 2.1896 2.3589 2.6853 2.9902 3.2694 1.50 1.5808 1.6218 1.6772 1.7054 1.7149 1.75 1.0847 1.0695 1.0323 0.9924 0.9538 0.25 1.2292 1.2121 1.1698 1.1244 1.0806 0.50 1.7570 1.8032 1.8634 1.8914 1.8980 0.75 2.3391 2.5247 2.8807 3.2106 3.5099 1.00 2.5301 2.8004 3.3819 4.0179 4.7085 1.25 2.0860 2.2438 2.5482 2.8330 3.0944 1.50 1.4546 1.4915 1.5427 1.5704 1.5816 1.75 0.9732 0.9609 0.9305 0.8975 0.8651
G. PRAKASH 834 Table 2. Relative efficiencthe estimator y for θ with respect under LLd δ. 0.25 .5 to * T F for n = 15 and different values of a, d an a δd 01 1.5 2 0.25 1 1 1.0109 0.9659 0.9182 .0624 .0484 0.50 1.2457 1.2660 1.2949 1.3099 1.3134 0.75 1.4014 1.4590 1.5717 1.6804 1.7841 1.00 1.4912 1.5691 1.7308 1.9004 2.0780 1.25 1.4755 1.5432 1.6762 1.8048 1.9277 1.50 1.3723 1.4048 1.4538 1.4835 1.4967 –0.05 –0.25 0.25 0.05 1.75 1.2137 1.2068 1.1766 1.1323 1.0812 0.25 1.0718 1.0584 1.0219 0.9777 0.9308 0.50 1.2612 1.2825 1.3130 1.3291 1.3335 0.75 1.4171 1.4759 1.5911 1.7021 1.8080 1.00 1.4875 1.5658 1.7285 1.8991 2.0777 1.25 1.4557 1.5196 1.6447 1.7654 1.8807 1.50 1.3262 1.3532 1.3930 1.4159 1.4246 1.75 1.1480 1.1375 1.1038 1.0595 1.0107 0.25 1.0895 1.0773 1.0428 1.0004 0.9548 0.50 1.2898 1.3132 1.3470 1.3656 1.3718 0.75 1.4427 1.5043 1.6247 1.7407 1.8513 1.00 1.4875 1.5645 1.7244 1.8920 2.0674 1.25 1.4012 1.4580 1.5689 1.6757 1.7775 1.50 1.2222 1.2406 1.2665 1.2796 1.2822 1.75 1.0133 0.9990 0.9631 0.9215 0.8782 0.25 1.0979 1.0862 1.0527 1.0112 0.9664 0.50 1.3031 1.3274 1.3630 1.3829 1.3900 0.75 1.4528 1.5158 1.6390 1.7577 1.8708 1.00 1.4803 1.5569 1.7158 1.8825 2.0568 1.25 1.3675 1.4210 1.5257 1.6263 1.7221 1.50 1.1660 1.1812 1.2021 1.2119 1.2127 1.75 0.9462 0.9313 0.8959 0.8566 0.8164 etter than an improved estiator in a wide of d R. S. Chhikara, “The Inverse Gaussian Distribution Its Statistiation—,” Journal of the Royal Statistical Society, Vol. 40, No. 3, b w me rangδ hich is defined here as the ratio between the true value and guess (prior point) value of the unknown parameter under the LLF. Thus, we suggest using the Minimax es timator under LLF for estimating the measure of disper sion under the Shrinkage setup. 7. References [1] J. L. Folks an andcal ApplicA Review 1978, pp. 263289. [2] J. L. Folks and R. S. Chhikara, “The Inverse Gaussian Distribution,” Marcel Dekker, New York, 1989. [3] V. Seshadri, “The Inverse Gaussian Distribution,” Statis tical Theory and Applications, Springer–Verlag, New York, 1998. [4] M. C. K. Tweedie, “Statistical Properties of Inverse Gau ssian Distribution–I,” The Annals of Mathematical Statis Copyright © 2011 SciRes. AM
G. PRAKASH 835 No. 2, 1957, pp. 362377. tics, Vol. 28, doi:10.1214/aoms/1177706964 [5] M. C. K. Tweedie, “Statistical Properties of Inverse Gaussian Distribution–II,” The Annals Statistics, Vol. 28, No. 2, 1957, p of Mathematical p. 696705. doi:10.1214/aoms/1177706881 [6] J. J. Schuster, “On the Inverse Gaussian Distribution,” Journal of American Statistical Association, V 324, 1968, pp. 15141516. ol. 63, No. .2307/2285899doi:10 Applied etrics and Statistics, Amsterdam ommunication in Sta [7] A. Parsian and S. N. U. A. Kirmani, “Estimation under LINEX Loss Function,” In: A. Ullah, A. T. K. Wan, A. Chaturvedi and M. Dekker, Eds., Handbook of Econometrics and Statistical Inference, CRC Press, Boca Raton, 2002, pp. 5376. [8] H. R. Varian, “A Bayesian Approach to Real Estate As sessment,” In: S. E. Feinberge and A. Zellner, Eds., Stud ies in Bayesian Econom North Holland, 1975, pp. 195208. [9] D. C. Singh, G. Prakash and P. Singh, “Shrinkage Testi mators for the Shape Parameter of Pareto Distribution Using the Linex Loss Function,” C tistics: Theory and Methods, Vol. 36, No. 4, 2007, pp. 741753. doi:10.1080/03610920601033694 [10] J. R. Thompson, “Some Shrinkage Techniques for Esti mating the Mean,” Journal of the American Statistical Association, Vol. 63, No. 321, 1968, pp. 113122. doi:10.2307/2283832 [11] B. N. Pandey and H. J. Malik, “Some Improved E tors for a Measure of stima Dispersion of an Inverse Gaussian Distribution,” Communications in Statistics: Theory and Methods, Vol. 17, 1988, pp. 39353949. doi:10.1080/03610928808829847 [12] G. Prakash and D. C. Singh, “Shrinkage the Inverse Dispersion of the Inve Testimators for rse Gaussian Distribu ,” Ph.D.  tion under the Linex Loss Function,” Austrian Journal of Statistics, Vol. 35, No. 4, 2006, pp. 463470. [13] T. Palmer, “Certain NonClassical Inference Procedures Applied to the Inverse Gaussian Distribution Dissertation, Oklahoma State University, Stillwater, 1973. [14] A. K. Banerjee and G. K.Bhattacharya, “Bayesian Results for the Inverse Gaussian Distribution with an Applica tion,” Technometrics, Vol. 21, No. 2, 1979, pp. 247251. doi:10.2307/1268523 [15] J. I. Hodge and E. L. Lehmann, “Some Problems in Minimax Estimation,” Annals of Mathematical Statistics, Vol. 21, No. 2, 1950, pp. 182197. doi:10.1214/aoms/1177729838 Copyright © 2011 SciRes. AM
